Let's fix a coordinates system $(x,y,z)$ with origin $O$. Considering a (screw or edge) dislocation and let the coordinate system $(x',y',z')$ with origin $O'$ move with the dislocation, and impose the condition that the two origins $O$ and $O'$ coincide at $t=0$. Then the following transformation holds
\begin{array}{l}x^{\prime}=\frac{x-v t}{\left(1-v^{2} / c_{t}^{2}\right)^{1 / 2}} \\ y^{\prime}=y \\ z^{\prime}=z \\ t^{\prime}=\frac{t-v x / c_{t}^{2}}{\left(1-v^{2} / c_{t}^{2}\right)^{1 / 2}}\end{array}
where $c_t$ is the speed of sound in the given material.
Does this transformation hold for a general dislocation (e.g. a loop dislocation) or what is the general transformation from the rest frame of the material (its lattice) to the moving frame of the dislocation?
